Cavitation
Cavitation: Theoretical Foundations
Overview
Professor, cavitation is the phenomenon where bubbles form in water, right?
Cavitation is a phenomenon where vapor cavities (cavities) form in a liquid when the local pressure of the liquid falls below its saturated vapor pressure. It occurs in high-speed flow fields such as pump impellers, ship propellers, and valve constrictions.
Is it different from boiling?
Boiling is driven by temperature increase, while cavitation is driven by pressure decrease. In cavitation, when bubbles move to a high-pressure region, they collapse rapidly, generating shock waves of thousands of atmospheres and localized high-temperature spots. This causes erosion.
Governing Equations
What models are used in CFD?
First, the cavitation number is the fundamental parameter.
In CFD, the homogeneous mixture model is mainstream, adding source terms to the transport equation for vapor volume fraction $\alpha_v$.
What types of source term models are there?
Let's compare typical cavitation models.
| Model | Basic Concept | Features |
|---|---|---|
| Schnerr-Sauer | Based on Rayleigh-Plesset equation | Bubble number density $n_0$ is a parameter |
| Zwart-Gerber-Belamri | Simplified RP equation | Fluent standard, controlled by adjustment coefficients |
| Singhal (Full Cavitation) | Mass transport | Considers non-condensable gas (dissolved air) |
| Kunz | Based on artificial compressibility | Suitable for steady-state calculations |
The underlying Rayleigh-Plesset equation describes the growth of a spherical bubble.
The 4th term is viscous damping, the 5th term is surface tension. In CFD models, the evaporation rate is derived from a simplified form ignoring the second-order and viscous terms.
How is the bubble number density $n_0$ determined?
For general water, $n_0 = 10^{13}$ /mยณ is the default value. In the Schnerr-Sauer model, the bubble radius is found using $R_B = \left(\frac{3\alpha_v}{4\pi n_0}\right)^{1/3}$, and the evaporation rate is calculated.
Bubbles That Break PropellersโThe Moment Cavitation Changed History
In 1893, the British destroyer "Daring" failed to reach its design speed, and mysterious damage kept occurring on its propeller. The phenomenon discovered by Osborne Reynolds and his successors investigating this was "cavitation." When local pressure falls below the vapor pressure of water (2.3 kPa at 20ยฐC), vapor bubbles form, and upon collapse, they generate shock pressures of hundreds of MPa, eroding the propeller material. This discovery led to the definition of the cavitation number ฯ = (p-pv)/(0.5ฯuยฒ), a dimensionless number that forms the foundation of modern hydraulic machinery design.
Computational Methods for Cavitation
Details of Numerical Methods
What are the numerical key points in cavitation analysis?
In vapor regions, the speed of sound of the mixture drastically decreases, making compressibility effects significant. The speed of sound for a water/vapor mixture can be much lower than that of pure water (approx. 1500 m/s), sometimes dropping to a few m/s.
Therefore, the pressure-density coupling is strong, and the Coupled algorithm is recommended for pressure-based solvers. Density-based solvers can sometimes be more stable.
Selection of Turbulence Models
Which turbulence model should I use?
The standard k-ฮต model overestimates turbulent viscosity and suppresses the unsteadiness of cavities. The Reboud correction is effective.
With $n \approx 10$, turbulent viscosity in the mixing region is reduced, allowing reproduction of cavity shedding. SST k-ฯ also shows good results in cavitation analysis.
To capture the details of unsteady cloud cavitation, DES, DDES, or LES is required.
Solver Settings
| Parameter | Recommended Value | Reason |
|---|---|---|
| Pressure-Velocity Coupling | Coupled | Strong pressure-density coupling |
| Spatial Discretization | 2nd order or higher | Resolution of cavity shape |
| Interface Courant Number | < 0.5 | Capturing bubble growth/collapse |
| Reference Pressure | Absolute pressure basis | For comparison with vapor pressure |
Implementation in OpenFOAM
Which solver is used in OpenFOAM?
interPhaseChangeFoam is the cavitation-capable VOF solver. Specify the model in constant/transportProperties. You can choose from SchnerrSauer, Kunz, and Merkle.
Settings in Fluent
Please also tell me about the key points on the Fluent side.
Select VOF or Mixture in the Multiphase Model and enable the Cavitation Model. Zwart-Gerber-Belamri is the default, with evaporation coefficient $F_{evap} = 50$ and condensation coefficient $F_{cond} = 0.01$ as standard values. The asymmetric coefficients reflect that collapse is more rapid than evaporation.
Schnerr-Sauer vs ZwartโThe Reality of Cavitation Model Selection
In CFD cavitation analysis, the choice of mass transport model is always a topic of discussion. The Schnerr-Sauer model rigorously derives the volume change of a single bubble from the Rayleigh equation and has the advantage of not requiring specification of nucleation density. On the other hand, the Zwart model is the Fluent standard with a wide track record and can reproduce hysteresis behavior through asymmetric evaporation/condensation coefficients (Ce=0.02, Cc=0.01). In pump inducer validation, even with the same mesh, the predicted cavitation inception ฯ can differ by more than 20% between the two models, making it dangerous to choose a model without matching experimental values.
Cavitation in Practice
Practical Guide
Please tell me the practical procedure for cavitation analysis.
Let me explain using a pump impeller analysis as an example.
1. Single-phase steady-state analysis: Fully converge the flow field without cavitation
2. Vapor pressure setting: Accurate saturated vapor pressure at operating temperature (3170 Pa for water at 25ยฐC)
3. Enable cavitation: Restart from the single-phase solution and enable the model
4. Gradual reduction of NPSH: Lower the inlet pressure to induce cavitation
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